In 2063, Zefram Cochrane launched the Phoenix and broke the light barrier by going to warp. Originally the physicist who had originally wanted to take humanity to the stars before the brutality of World War 3 made him realise that maybe it would be better if humanity did not go out into space, as they would be more liable to blow it all to kingdom come as they had nearly done on their own world. As he had worked on the Phoenix with what was left over from his team, his dream was coloured with cynicism and he decided to sell it off to the Indonesian Space Agency, knowing they still existed and they might make something better out of it while he just went off on his own to live a quiet life.
But something changed in Cochrane. There were those stories he spread about cybernetic creatures who'd time travelled back from the future to enslave the human race in the past, but nobody took him seriously enough and merely thought he had gotten over his depression. But whatever it was, he enthusiastically piloted the Phoenix himself, broke the light barrier, and encountered the Vulcans.
The discovery of alien life shattered so many preconceptions and suddenly everything changed for the better. Space exploration took priority next to the rebuilding of the Earth because it was hoped if they could start a colony quickly, they could ensure if something went wrong, at least a few thousand humans would survive. Fortunately, such a disaster never came.
While they provided help, the Vulcans were taking a backstreet. Deep down they were both impressed and worried and even terrified by how quickly humanity took their lessons. In less than half a century, poverty, disease, and war were virtually wiped away. Political ideologies like the ones that saw the rise of the Nazis in the mid-20th century and the ECON's genocidal aims just vanished while resources went into rebuilding Earth and looking for new worlds to colonise to help in the rebuilding. From a certain perspective, many Vulcans could understand the humans' point of view; their world was devastated and their civilisation had come close to being wiped out and they wanted to expend their resources on making sure it never happened again.
As other bodies in space were colonised, a wave of explorers left Earth and began looking for new resources. Warp drive ships and probes were sent out into space to help, old astronauts who'd survived the war began training the next generations while they worked to understand their new technologies. They had artificial gravity now, and faster than light, and soon there were asteroid colonies and colonies on Mars and on the moon. But those were all in the solar system. There weren't any planets beyond. That was why, when it was discovered, the planet that was so much like Earth was so valued.
Dubbed Terra Nova, the planet was to become the Great Experiment to see if it was possible for humans to migrate beyond Earth and survive the trip while colonising a potentially dangerous world. During the construction process of building an interstellar starship, a new faster-than-light drive was being pioneered.
The Queller drive was a faster-than-light engine similar to a warp drive, only it folded space and time rather than warped it, and its equations could be implemented into a conventional warp drive. Once it was tested and developed, the Queller drive was instantly accepted by Zefram Cochrane, who rather than be hurt or annoyed, was delighted they had a more powerful engine, and he began working with the scientist Hans Queller to develop the technology even more.
Within two years, the partnership paid off and soon a small fleet of ships left Earth and travelled to Terra Nova. They reached the planet in a month, rather than the ten years it was originally believed it would take, and along the way, they collected and sent back vital and valuable data for scientists on Earth, but some of the fleet would branch away from Terra Nova, and embark on a short term exploration mission before making the return home.
The Queller-Cochrane drive became instrumental in understanding the mystery behind the sudden loss of contact with the Terra Nova colony helped save them from radiation poisoning, and allowed them to further develop the planet. From that point forward, the colony began to grow and it was not the only one as fresh colonies developed and grew on many worlds.
Without the Queller drive, the Cochrane warp drive had meant exploration missions would spend years travelling the stars in waves of short-range starships, sending information back to the mother planet. With the Queller drive, longer-term interstellar travel expeditions became possible. After the war, many astronauts had been collateral damage during the fighting, or they had died later on. But now some of the surviving astronauts were crewing those ships, and training the next generation.
However, advancements in the Queller-Cochrane drive increased all the time. Building starships that could go much further and faster was the priority, and after Queller's drive saw the linking between the two technologies, many propulsion scientists with their own ideas came onto the scene. But the Queller-Cochrane engine ruled supreme. But the newly inaugurated Starfleet, an amalgamation of the surviving space exploration and defence agencies and organisations, merged together to form one organisation, with all the backing of the United Earth government. Soon other warp physicists like Henry Archer came in and began working on the Warp 5 enginewhich would go out even further into space. Soon, humanity cracked the problems with Warp 2 and Warp 3, and new starships and freighters were sent out on extended missions, to explore strange new worlds and to explore space.
The Warp Four breakthrough changed things, though, and the Cochrane class starships were launched. Armed with transmitters beaming information back home, and with their unattached crews in suspended animation, they were to spend years out in space for extended periods of time before the keels of the first warp 5 ships were laid down, beginning a legacy that would last generations...
Authors Note, I got the Queller drive out of Space 1999. It was a nuclear engine that was lethal, but it allowed space ships to traverse interstellar space. I felt it was appropriate.